Let's solve the composite functions step by step based on the provided information.

We are given the following functions:

• $f(x) = x^5$
• $g(x) = \sin(x)$

1. $f(g(x))$

The composite function $f(g(x))$ means applying $g(x)$ first, and then applying $f(x)$.

$f(g(x)) = f(\sin(x)) = (\sin(x))^5 = \sin^5(x)$

So, the answer for $f(g(x))$ is:

$f(g(x)) = \sin^5(x)$

2. $g(f(x))$

The composite function $g(f(x))$ means applying $f(x)$ first, and then applying $g(x)$.

$g(f(x)) = g(x^5) = \sin(x^5)$

So, the answer for $g(f(x))$ is:

$g(f(x)) = \sin(x^5)$

Would you like further details or explanations on any part of this process?

Here are some related questions to expand on this topic:

1. How do you find the derivative of composite functions like $f(g(x))$?
2. What is the chain rule, and how is it applied in differentiating composite functions?
3. How do you calculate the inverse of composite functions?
4. Can you simplify $f(g(x))$ for more complex functions, like trigonometric compositions?
5. How do composite functions apply in real-world scenarios, such as physics or engineering?

Tip: When working with composite functions, it's useful to think of each function as a transformation applied in a specific sequence. This helps keep track of which function to apply first.